The following questions have some relevance to irrational numbers from a historical perspective. If you find information that is useful you can post it here or link to it. Describe in detail how you found the information. Sign your posts with 4 tildes to create a link to your name for messaging purposes.
1. What is an irrational number by definition? Give an example of an irrational number (I expect everyone who logs onto this page to do this. You can put a four tilde signature after your example.)
According to www.dictionary.com, an irrational number is "any real number that cannot be expressed as a ratio between two integers." Another source on dictionary.com stated that "if written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without repetition." Examples of this would be pi and the square root of 2, because like the definition specifies, these two numbers have non-repeating, infinite digits after the decimal point. - herecomesthesun Nov 1, 2007
According to www.math.com, an irrational number is "A number that cannot be expressed as the ratio of two integers." An example of an irrational number would be the square root of 6 since it can't be expressed without a never ending decimal. - linz16 Nov 1, 2007
According to http://mathworld.wolfram.com ,
An irrational number is "a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.
The most famous irrational number is the square root of 2, sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of the square root while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include the square root of three,pie, etc." - heyyyk Nov 4, 2007
2. Who was Pythagoras?
Pythagoras was a philosopher. he came up with the Pythagorean theorem, he also founded a religious movement called Pythagoreanism. Also Pythagoras believed that all numbers had perfect squares. [[user:waterpolochick1|1193943149]--------(IMPeachy)The theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.(http://who2.com/ask/pythagoras.html)
3. Who were the Pythagoreans? What did they believe?
I searched Google, typed in "Pythagoreans" and found out that "the Pythagoreans believed (but failed to prove) that the universe could be understood in terms of whole numbers. This belief stemmed from observations in music, mathematics and astronomy. For example, they noticed that vibrating strings produce harmonious tones when the ratios of their lengths are whole numbers." This is from: this site. Another site explained that they attended "The Pythagorean School of Pythagoras which was every bit as much a religious school as of a school of mathematics." Which I found at this site.- herecomesthesun Nov 1, 2007
Mr. Vizza had brought this to my attention from the second website I had cited above (^), and I think it is very interesting. In Pythagorean philosophy..."All things of the universe had a numerical attribute that uniquely described them. For example,
The number one : the number of reason.
The number two: the first even or female number, the number of opinion.
The number three: the first true male number, the number of harmony.
The number four: the number of justice or retribution.
The number five: marriage.
The number six: creation
The number ten: the tetractys, the number of the universe." - herecomesthesun Nov 5, 2007
4. Try to find reference to a story in which someone was executed for challenging the beliefs of the Pythagoreans. On the discussion page, start or join a discussion on your thoughts on this story.
A philosopher named Hippasus was the man who oppesed Pythagoras idea that all numbers are perfect square. Hippasus was trying to write the square route of two as a decimal and could not. In the story it was said that he discovered this at sea and was thrown over board by the Pythagoreans. - waterpolochick1 Nov 1, 2007
I went on this website http://math.about.com/library/blpyth.htm , and found out that...
"Hippasus of Metapontum was born in the 5th century BC. It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn't believe there was a need for any other numbers.
The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive and did not want their discoveries to 'get out' so to speak. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio. Hence, the Pythagorean Theorem which crushed their original beliefs. Thus, they certainly didn't want Hippasus' discovery to be revealed and shatter their pride and core beliefs.
Unfortunately, very little is known about the Pythagoreans due to their very secretive ways. However, there are many stories about what happened to Hippasus as a result of his discovery. Some say that he was drowned, most believe that he was expelled from the Pythagorean Society and many believe he was killed and his discovery with stood the test of time." - heyyyk Nov 4, 2007
1. What is an irrational number by definition? Give an example of an irrational number (I expect everyone who logs onto this page to do this. You can put a four tilde signature after your example.)
According to www.dictionary.com, an irrational number is "any real number that cannot be expressed as a ratio between two integers." Another source on dictionary.com stated that "if written in decimal notation, an irrational number would have an infinite number of digits to the right of the decimal point, without repetition." Examples of this would be pi and the square root of 2, because like the definition specifies, these two numbers have non-repeating, infinite digits after the decimal point. -
According to www.math.com, an irrational number is "A number that cannot be expressed as the ratio of two integers." An example of an irrational number would be the square root of 6 since it can't be expressed without a never ending decimal. -
According to http://mathworld.wolfram.com ,
An irrational number is "a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.
The most famous irrational number is the square root of 2, sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of the square root while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include the square root of three,pie, etc." -
2. Who was Pythagoras?
Pythagoras was a philosopher. he came up with the Pythagorean theorem, he also founded a religious movement called Pythagoreanism. Also Pythagoras believed that all numbers had perfect squares. [[user:waterpolochick1|1193943149]--------(IMPeachy)The theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.(http://who2.com/ask/pythagoras.html)
(http://www.frontiernet.net/~imaging/pythagorean.html) -
3. Who were the Pythagoreans? What did they believe?
I searched Google, typed in "Pythagoreans" and found out that "the Pythagoreans believed (but failed to prove) that the universe could be understood in terms of whole numbers. This belief stemmed from observations in music, mathematics and astronomy. For example, they noticed that vibrating strings produce harmonious tones when the ratios of their lengths are whole numbers." This is from: this site. Another site explained that they attended "The Pythagorean School of Pythagoras which was every bit as much a religious school as of a school of mathematics." Which I found at this site. -
Mr. Vizza had brought this to my attention from the second website I had cited above (^), and I think it is very interesting. In Pythagorean philosophy..."All things of the universe had a numerical attribute that uniquely described them. For example,
4. Try to find reference to a story in which someone was executed for challenging the beliefs of the Pythagoreans. On the discussion page, start or join a discussion on your thoughts on this story.
A philosopher named Hippasus was the man who oppesed Pythagoras idea that all numbers are perfect square. Hippasus was trying to write the square route of two as a decimal and could not. In the story it was said that he discovered this at sea and was thrown over board by the Pythagoreans. -
I went on this website http://math.about.com/library/blpyth.htm , and found out that...
"Hippasus of Metapontum was born in the 5th century BC. It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn't believe there was a need for any other numbers.
The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive and did not want their discoveries to 'get out' so to speak. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio. Hence, the Pythagorean Theorem which crushed their original beliefs. Thus, they certainly didn't want Hippasus' discovery to be revealed and shatter their pride and core beliefs.
Unfortunately, very little is known about the Pythagoreans due to their very secretive ways. However, there are many stories about what happened to Hippasus as a result of his discovery. Some say that he was drowned, most believe that he was expelled from the Pythagorean Society and many believe he was killed and his discovery with stood the test of time."
-